In each of the following, find the point of intersection of...

In each of the following, find the point of intersection of the two given lines, and if they do not intersect, explain why. a) $L_{1}: r=(2,2,3)+t(1,3,1)$ $L_{2}: r=(2,3,4)+t(1,4,2)$ b) $L_{1}: r=(-1,3,1)+t(4,1,0)$ $L_{2}: \mathbf{r}=(-13,1,2)+\mathrm{f}(12,6,3)$ c) $L_{1}: r=(1,3,5)+t(7,1,-3)$ $L_{2}: \mathbf{r}=(4,6,7)+t(-1,0,2)$ d) $L_{1}:\left(\begin{array}{l}x \\ y \\ z\end{array}\right)=\left(\begin{array}{l}3 \\ 4 \\ 6\end{array}\right)+t\left(\begin{array}{r}-2 \\ 1 \\ -1\end{array}\right)$ $L_{2}:\left(\begin{array}{l}x \\ y \\ z\end{array}\right)=\left(\begin{array}{r}5 \\ -2 \\ 7\end{array}\right)+s\left(\begin{array}{r}-4 \\ 2 \\ -2\end{array}\right)$

In each of the following, find the point of intersection of the two given lines, and if they do not intersect, explain why.
a) $L_{1}: r=(2,2,3)+t(1,3,1)$
$L_{2}: r=(2,3,4)+t(1,4,2)$
b) $L_{1}: r=(-1,3,1)+t(4,1,0)$
$L_{2}: \mathbf{r}=(-13,1,2)+\mathrm{f}(12,6,3)$
c) $L_{1}: r=(1,3,5)+t(7,1,-3)$
$L_{2}: \mathbf{r}=(4,6,7)+t(-1,0,2)$
d) $L_{1}:\left(\begin{array}{l}x \\ y \\ z\end{array}\right)=\left(\begin{array}{l}3 \\ 4 \\ 6\end{array}\right)+t\left(\begin{array}{r}-2 \\ 1 \\ -1\end{array}\right)$
$L_{2}:\left(\begin{array}{l}x \\ y \\ z\end{array}\right)=\left(\begin{array}{r}5 \\ -2 \\ 7\end{array}\right)+s\left(\begin{array}{r}-4 \\ 2 \\ -2\end{array}\right)$

Key Concepts

Consistency and Special Cases

In the context of line intersections, the obtained system of equations can have a unique solution (indicating a single intersection point), no solution (indicating that the lines are either parallel and distinct or skew), or infinitely many solutions (indicating that the lines are coincident). Checking the consistency of this system determines the relationship between the lines.

System of Linear Equations

Equating the corresponding components of two parametric line equations results in a system of linear equations. Solving this system is crucial as it provides the parameter values that, when substituted back into either line equation, yield the intersection point if one exists.

Parametric Equations of Lines

Parametric equations represent a line in space by expressing the coordinates of any point on the line as functions of a parameter. The general form is a fixed point (the position vector) plus the parameter multiplied by a direction vector, which clearly indicates the line’s trajectory in space.

Intersection of Lines

The intersection point of two lines is a point that lies on both lines simultaneously. To find such a point, we set the parametric equations equal to each other, which translates into a set of equations for the parameters. A solution exists if and only if there is a common point that satisfies both sets of equations.

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